Markov subshifts and partial representation of F_n

$C^*$-algebras arising from Dyck systems of topological Markov chains

Let $A$ be an $N \times N $ irreducible matrix with entries in $\{0,1\}$. We define the topological Markov Dyck shift $D_A$ to be a nonsofic subshift consisting of the $2N$ brackets $(_1,...,(_N,)_1,...,)_N$ with both standard bracket rule and Markov chain rule coming from $A$. The subshift is regarded as a subshift defined by the canonical generators $S_1^*,..., S_N^*, S_1,..., S_N $ of the Cuntz-Krieger algebra ${\Cal O}_A$. We construct an irreducible $λ$-graph system ${{\frak L}^{Ch(D_A)}}$ that presents the subshift $D_A$ so that we have an associated simple purely infinite $C^*$-algebra …

Higher rank graphs, k-subshifts and k-automata

Given a $k$-graph $Λ$ we construct a Markov space $M_Λ$, and a collection of $k$ pairwise commuting cellular automata on $M_Λ$, providing for a factorization of Markov's shift. Iterating these maps we obtain an action of ${\mathbb N}^k$ on $M_Λ$ which is then used to form a semidirect product groupoid $M_Λ\rtimes {\mathbb N}^k$. This groupoid turns out to be identical to the path groupoid constructed by Kumjian and Pask, and hence its C*-algebra is isomorphic to the higher rank graph C*-algebra of $Λ$.

带有关系的部分群代数的动态结构，及其在子移位代数中的应用

The dynamical structure of partial group algebras with relations, with applications to subshift algebras

本文研究了带有关系的部分群代数的动态结构，并展示了如何通过拓扑部分作用将子移位代数表示为部分群代数，同时探讨了其简洁性与动力学的关系。

Partial actions and subshifts

Given a finite alphabet $Λ$, and a not necessarily finite type subshift $X\subseteq Λ^\infty$, we introduce a partial action of the free group $F(Λ)$ on a certain compactification $Ω_X$ of $X$, which we call the spectral partial action. The space $Ω_X$ has already appeared in many papers in the subject, arising as the spectrum of a commutative C*-algebra usually denoted by ${\cal D}_X$. Since the descriptions given of $Ω_X$ in the literature are often somewhat terse and obscure, one of our main goals is to present a sensible model for it which allows for a detailed study of its structure, as w…

Convex subshifts, separated Bratteli diagrams, and ideal structure of tame separated graph algebras

We introduce a new class of partial actions of free groups on totally disconnected compact Hausdorff spaces, which we call convex subshifts. These serve as an abstract framework for the partial actions associated with finite separated graphs in much the same way as classical subshifts generalize the edge shift of a finite graph. We define the notion of a finite type convex subshift and show that any such subshift is Kakutani equivalent to the partial action associated with a finite bipartite separated graph. We then study the ideal structure of both the full and the reduced tame graph C*-algeb…

Markovianity and the Thompson Monoid $F^+$

We introduce a new distributional invariance principle, called `partial spreadability', which emerges from the representation theory of the Thompson monoid $F^+$ in noncommutative probability spaces. We show that a partially spreadable sequence of noncommutative random variables is adapted to a local Markov filtration. Conversely we show that a large class of noncommutative stationary Markov sequences provides representations of the Thompson monoid $F^+$. In the particular case of a classical probability space, we arrive at a de Finetti theorem for stationary Markov sequences with values in a …